b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 11.1%

Time: 47.5s

Alternatives: 3

Speedup: 9.9×

• could not determine a ground truth

  1. a = 8.469987589689056e-293
  2. b = 2.8062255669028243e+279
  3. angle = 5.085999769410727e-186
  4. x-scale = 8.341913633762392e-128
  5. y-scale = 1.2329283825690734e+302

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) - \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (-
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) - sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 0.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) - \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (-
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) - sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 11.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \left(a\_m \cdot b\_m\right) \cdot 4\\ t_1 := \left(-a\_m\right) \cdot b\_m\\ t_2 := \frac{t\_0}{y-scale \cdot x-scale\_m} \cdot \frac{t\_1}{y-scale \cdot x-scale\_m}\\ t_3 := \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_4 := 0.5 + 0.5 \cdot t\_3\\ t_5 := \left(t\_1 \cdot b\_m\right) \cdot a\_m\\ \mathbf{if}\;a\_m \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot 0}}{t\_2}\\ \mathbf{elif}\;a\_m \leq 7.1 \cdot 10^{+74}:\\ \;\;\;\;\left(0.25 \cdot \frac{b\_m \cdot \left(y-scale \cdot \sqrt{-8 \cdot \frac{{a\_m}^{4} \cdot \left(\left(\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + 0.5 \cdot \frac{t\_3}{{y-scale}^{2}}\right) - 0.5 \cdot \frac{1}{{y-scale}^{2}}\right)}{{y-scale}^{2}}}\right)}{{a\_m}^{2}}\right) \cdot \left(y-scale \cdot x-scale\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{\left(8 \cdot t\_5\right) \cdot \left(t\_5 \cdot \frac{{b\_m}^{2} \cdot t\_4 - \sqrt{{b\_m}^{4} \cdot {t\_4}^{2}}}{{x-scale\_m}^{2}}\right)}}{\left|y-scale \cdot x-scale\_m\right|}}{t\_0 \cdot \left(a\_m \cdot b\_m\right)} \cdot \left(y-scale \cdot x-scale\_m\right)\right) \cdot \left(y-scale \cdot x-scale\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* (* a_m b_m) 4.0))
        (t_1 (* (- a_m) b_m))
        (t_2 (* (/ t_0 (* y-scale x-scale_m)) (/ t_1 (* y-scale x-scale_m))))
        (t_3 (cos (* 0.011111111111111112 (* angle PI))))
        (t_4 (+ 0.5 (* 0.5 t_3)))
        (t_5 (* (* t_1 b_m) a_m)))
   (if (<= a_m 2e-167)
     (/ (- (sqrt (* (* (* 2.0 t_2) (* (* b_m a_m) (* b_m (- a_m)))) 0.0))) t_2)
     (if (<= a_m 7.1e+74)
       (*
        (*
         0.25
         (/
          (*
           b_m
           (*
            y-scale
            (sqrt
             (*
              -8.0
              (/
               (*
                (pow a_m 4.0)
                (-
                 (+
                  (sqrt
                   (/
                    (pow (sin (* 0.005555555555555556 (* angle PI))) 4.0)
                    (pow y-scale 4.0)))
                  (* 0.5 (/ t_3 (pow y-scale 2.0))))
                 (* 0.5 (/ 1.0 (pow y-scale 2.0)))))
               (pow y-scale 2.0))))))
          (pow a_m 2.0)))
        (* y-scale x-scale_m))
       (*
        (*
         (/
          (/
           (sqrt
            (*
             (* 8.0 t_5)
             (*
              t_5
              (/
               (- (* (pow b_m 2.0) t_4) (sqrt (* (pow b_m 4.0) (pow t_4 2.0))))
               (pow x-scale_m 2.0)))))
           (fabs (* y-scale x-scale_m)))
          (* t_0 (* a_m b_m)))
         (* y-scale x-scale_m))
        (* y-scale x-scale_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (a_m * b_m) * 4.0;
	double t_1 = -a_m * b_m;
	double t_2 = (t_0 / (y_45_scale * x_45_scale_m)) * (t_1 / (y_45_scale * x_45_scale_m));
	double t_3 = cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_4 = 0.5 + (0.5 * t_3);
	double t_5 = (t_1 * b_m) * a_m;
	double tmp;
	if (a_m <= 2e-167) {
		tmp = -sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_2;
	} else if (a_m <= 7.1e+74) {
		tmp = (0.25 * ((b_m * (y_45_scale * sqrt((-8.0 * ((pow(a_m, 4.0) * ((sqrt((pow(sin((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0) / pow(y_45_scale, 4.0))) + (0.5 * (t_3 / pow(y_45_scale, 2.0)))) - (0.5 * (1.0 / pow(y_45_scale, 2.0))))) / pow(y_45_scale, 2.0)))))) / pow(a_m, 2.0))) * (y_45_scale * x_45_scale_m);
	} else {
		tmp = (((sqrt(((8.0 * t_5) * (t_5 * (((pow(b_m, 2.0) * t_4) - sqrt((pow(b_m, 4.0) * pow(t_4, 2.0)))) / pow(x_45_scale_m, 2.0))))) / fabs((y_45_scale * x_45_scale_m))) / (t_0 * (a_m * b_m))) * (y_45_scale * x_45_scale_m)) * (y_45_scale * x_45_scale_m);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (a_m * b_m) * 4.0;
	double t_1 = -a_m * b_m;
	double t_2 = (t_0 / (y_45_scale * x_45_scale_m)) * (t_1 / (y_45_scale * x_45_scale_m));
	double t_3 = Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double t_4 = 0.5 + (0.5 * t_3);
	double t_5 = (t_1 * b_m) * a_m;
	double tmp;
	if (a_m <= 2e-167) {
		tmp = -Math.sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_2;
	} else if (a_m <= 7.1e+74) {
		tmp = (0.25 * ((b_m * (y_45_scale * Math.sqrt((-8.0 * ((Math.pow(a_m, 4.0) * ((Math.sqrt((Math.pow(Math.sin((0.005555555555555556 * (angle * Math.PI))), 4.0) / Math.pow(y_45_scale, 4.0))) + (0.5 * (t_3 / Math.pow(y_45_scale, 2.0)))) - (0.5 * (1.0 / Math.pow(y_45_scale, 2.0))))) / Math.pow(y_45_scale, 2.0)))))) / Math.pow(a_m, 2.0))) * (y_45_scale * x_45_scale_m);
	} else {
		tmp = (((Math.sqrt(((8.0 * t_5) * (t_5 * (((Math.pow(b_m, 2.0) * t_4) - Math.sqrt((Math.pow(b_m, 4.0) * Math.pow(t_4, 2.0)))) / Math.pow(x_45_scale_m, 2.0))))) / Math.abs((y_45_scale * x_45_scale_m))) / (t_0 * (a_m * b_m))) * (y_45_scale * x_45_scale_m)) * (y_45_scale * x_45_scale_m);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	t_0 = (a_m * b_m) * 4.0
	t_1 = -a_m * b_m
	t_2 = (t_0 / (y_45_scale * x_45_scale_m)) * (t_1 / (y_45_scale * x_45_scale_m))
	t_3 = math.cos((0.011111111111111112 * (angle * math.pi)))
	t_4 = 0.5 + (0.5 * t_3)
	t_5 = (t_1 * b_m) * a_m
	tmp = 0
	if a_m <= 2e-167:
		tmp = -math.sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_2
	elif a_m <= 7.1e+74:
		tmp = (0.25 * ((b_m * (y_45_scale * math.sqrt((-8.0 * ((math.pow(a_m, 4.0) * ((math.sqrt((math.pow(math.sin((0.005555555555555556 * (angle * math.pi))), 4.0) / math.pow(y_45_scale, 4.0))) + (0.5 * (t_3 / math.pow(y_45_scale, 2.0)))) - (0.5 * (1.0 / math.pow(y_45_scale, 2.0))))) / math.pow(y_45_scale, 2.0)))))) / math.pow(a_m, 2.0))) * (y_45_scale * x_45_scale_m)
	else:
		tmp = (((math.sqrt(((8.0 * t_5) * (t_5 * (((math.pow(b_m, 2.0) * t_4) - math.sqrt((math.pow(b_m, 4.0) * math.pow(t_4, 2.0)))) / math.pow(x_45_scale_m, 2.0))))) / math.fabs((y_45_scale * x_45_scale_m))) / (t_0 * (a_m * b_m))) * (y_45_scale * x_45_scale_m)) * (y_45_scale * x_45_scale_m)
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(a_m * b_m) * 4.0)
	t_1 = Float64(Float64(-a_m) * b_m)
	t_2 = Float64(Float64(t_0 / Float64(y_45_scale * x_45_scale_m)) * Float64(t_1 / Float64(y_45_scale * x_45_scale_m)))
	t_3 = cos(Float64(0.011111111111111112 * Float64(angle * pi)))
	t_4 = Float64(0.5 + Float64(0.5 * t_3))
	t_5 = Float64(Float64(t_1 * b_m) * a_m)
	tmp = 0.0
	if (a_m <= 2e-167)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * 0.0))) / t_2);
	elseif (a_m <= 7.1e+74)
		tmp = Float64(Float64(0.25 * Float64(Float64(b_m * Float64(y_45_scale * sqrt(Float64(-8.0 * Float64(Float64((a_m ^ 4.0) * Float64(Float64(sqrt(Float64((sin(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0) / (y_45_scale ^ 4.0))) + Float64(0.5 * Float64(t_3 / (y_45_scale ^ 2.0)))) - Float64(0.5 * Float64(1.0 / (y_45_scale ^ 2.0))))) / (y_45_scale ^ 2.0)))))) / (a_m ^ 2.0))) * Float64(y_45_scale * x_45_scale_m));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * t_5) * Float64(t_5 * Float64(Float64(Float64((b_m ^ 2.0) * t_4) - sqrt(Float64((b_m ^ 4.0) * (t_4 ^ 2.0)))) / (x_45_scale_m ^ 2.0))))) / abs(Float64(y_45_scale * x_45_scale_m))) / Float64(t_0 * Float64(a_m * b_m))) * Float64(y_45_scale * x_45_scale_m)) * Float64(y_45_scale * x_45_scale_m));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = (a_m * b_m) * 4.0;
	t_1 = -a_m * b_m;
	t_2 = (t_0 / (y_45_scale * x_45_scale_m)) * (t_1 / (y_45_scale * x_45_scale_m));
	t_3 = cos((0.011111111111111112 * (angle * pi)));
	t_4 = 0.5 + (0.5 * t_3);
	t_5 = (t_1 * b_m) * a_m;
	tmp = 0.0;
	if (a_m <= 2e-167)
		tmp = -sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_2;
	elseif (a_m <= 7.1e+74)
		tmp = (0.25 * ((b_m * (y_45_scale * sqrt((-8.0 * (((a_m ^ 4.0) * ((sqrt(((sin((0.005555555555555556 * (angle * pi))) ^ 4.0) / (y_45_scale ^ 4.0))) + (0.5 * (t_3 / (y_45_scale ^ 2.0)))) - (0.5 * (1.0 / (y_45_scale ^ 2.0))))) / (y_45_scale ^ 2.0)))))) / (a_m ^ 2.0))) * (y_45_scale * x_45_scale_m);
	else
		tmp = (((sqrt(((8.0 * t_5) * (t_5 * ((((b_m ^ 2.0) * t_4) - sqrt(((b_m ^ 4.0) * (t_4 ^ 2.0)))) / (x_45_scale_m ^ 2.0))))) / abs((y_45_scale * x_45_scale_m))) / (t_0 * (a_m * b_m))) * (y_45_scale * x_45_scale_m)) * (y_45_scale * x_45_scale_m);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(a$95$m * b$95$m), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[((-a$95$m) * b$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(0.5 + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$1 * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision]}, If[LessEqual[a$95$m, 2e-167], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[a$95$m, 7.1e+74], N[(N[(0.25 * N[(N[(b$95$m * N[(y$45$scale * N[Sqrt[N[(-8.0 * N[(N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(N[Sqrt[N[(N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[y$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(t$95$3 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(N[(8.0 * t$95$5), $MachinePrecision] * N[(t$95$5 * N[(N[(N[(N[Power[b$95$m, 2.0], $MachinePrecision] * t$95$4), $MachinePrecision] - N[Sqrt[N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 2e-167

   1. Initial program 0.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 0.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(b \cdot a\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      14. pow2 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      15. times-frac N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale}} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      18. lower-/.f64 0.2

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot \left(-a\right)}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      20. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      21. lift-*.f64 0.2

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 0.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(b \cdot a\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}} \]

      14. pow2 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}} \]

      15. times-frac N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale}} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}} \]

      18. lower-/.f64 0.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot \left(-a\right)}}{y-scale \cdot x-scale}} \]

      20. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}} \]

      21. lift-*.f64 0.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}} \]

   8. Applied rewrites 0.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}}} \]

   9. Taylor expanded in y-scale around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{\color{blue}{{y-scale}^{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{\color{blue}{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       3. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       6. lower-pow.f64 4.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   11. Applied rewrites 4.4%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{\color{blue}{{y-scale}^{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   12. Taylor expanded in a around 0

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
   13. Step-by-step derivation

   14. Applied rewrites 9.6%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   if 2e-167 < a < 7.10000000000000002e74

   1. Initial program 0.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in x-scale around inf

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{\color{blue}{{\left(\frac{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{\color{blue}{2}}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   4. Applied rewrites 0.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{\color{blue}{{\left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 0.3%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 8\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot \left(a \cdot \left(\left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, -0.5\right), b \cdot b, \left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), -0.5\right) \cdot a\right) \cdot a\right)}{y-scale \cdot y-scale}\right) - \sqrt{{\left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5\right) \cdot b, b, \left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot a\right) \cdot a\right)}{y-scale \cdot y-scale}\right)}^{2}}\right)\right)\right)}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   7. Taylor expanded in x-scale around -inf

      \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \frac{y-scale \cdot \sqrt{-8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) - \frac{1}{2}\right)}{{y-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) - \frac{1}{2}\right)}{{y-scale}^{2}}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}\right)} \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 0.9%

      \[\leadsto \color{blue}{\left(-0.25 \cdot \frac{y-scale \cdot \sqrt{-8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(-0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) - 0.5\right)}{{y-scale}^{2}} + \frac{{b}^{2} \cdot \left(0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) - 0.5\right)}{{y-scale}^{2}}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}\right)} \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in b around -inf

       \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\frac{b \cdot \left(y-scale \cdot \sqrt{-8 \cdot \frac{{a}^{4} \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{y-scale}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{{y-scale}^{2}}\right)}{{y-scale}^{2}}}\right)}{{a}^{2}}}\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 4.1%

       \[\leadsto \left(0.25 \cdot \color{blue}{\frac{b \cdot \left(y-scale \cdot \sqrt{-8 \cdot \frac{{a}^{4} \cdot \left(\left(\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right) - 0.5 \cdot \frac{1}{{y-scale}^{2}}\right)}{{y-scale}^{2}}}\right)}{{a}^{2}}}\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 7.10000000000000002e74 < a

   1. Initial program 0.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Applied rewrites 0.4%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\left(\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right) - \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\left(\left(\frac{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{x-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}\right) - \sqrt{\frac{{b}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{x-scale}^{2}} - \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}}\right)}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 1.5%

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\left(\left(\frac{{b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} + \frac{{b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right) - \sqrt{\frac{{b}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}}\right)}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in x-scale around 0

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) - \sqrt{{b}^{4} \cdot {\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}}}{\color{blue}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) - \sqrt{{b}^{4} \cdot {\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}}}{{x-scale}^{\color{blue}{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 4.7%

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{{b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{b}^{4} \cdot {\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}{\color{blue}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 9.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \left(-a\_m\right) \cdot b\_m\\ t_1 := \frac{\left(a\_m \cdot b\_m\right) \cdot 4}{y-scale \cdot x-scale\_m} \cdot \frac{t\_0}{y-scale \cdot x-scale\_m}\\ \mathbf{if}\;y-scale \leq 7.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot 0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\frac{a\_m \cdot a\_m - \sqrt{{a\_m}^{4}}}{y-scale \cdot y-scale} \cdot \left(\left(\frac{\left(b\_m \cdot a\_m\right) \cdot \left(-4 \cdot \left(b\_m \cdot a\_m\right)\right)}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\_m\right) \cdot x-scale\_m} \cdot 2\right) \cdot \left(\left(\left(-a\_m\right) \cdot a\_m\right) \cdot \left(b\_m \cdot b\_m\right)\right)\right)}}{4 \cdot \left(b\_m \cdot a\_m\right)} \cdot \left(x-scale\_m \cdot y-scale\right)}{t\_0} \cdot \left(x-scale\_m \cdot y-scale\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* (- a_m) b_m))
        (t_1
         (*
          (/ (* (* a_m b_m) 4.0) (* y-scale x-scale_m))
          (/ t_0 (* y-scale x-scale_m)))))
   (if (<= y-scale 7.8e+105)
     (/ (- (sqrt (* (* (* 2.0 t_1) (* (* b_m a_m) (* b_m (- a_m)))) 0.0))) t_1)
     (*
      (/
       (*
        (/
         (-
          (sqrt
           (*
            (/ (- (* a_m a_m) (sqrt (pow a_m 4.0))) (* y-scale y-scale))
            (*
             (*
              (/
               (* (* b_m a_m) (* -4.0 (* b_m a_m)))
               (* (* (* y-scale y-scale) x-scale_m) x-scale_m))
              2.0)
             (* (* (- a_m) a_m) (* b_m b_m))))))
         (* 4.0 (* b_m a_m)))
        (* x-scale_m y-scale))
       t_0)
      (* x-scale_m y-scale)))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = -a_m * b_m;
	double t_1 = (((a_m * b_m) * 4.0) / (y_45_scale * x_45_scale_m)) * (t_0 / (y_45_scale * x_45_scale_m));
	double tmp;
	if (y_45_scale <= 7.8e+105) {
		tmp = -sqrt((((2.0 * t_1) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_1;
	} else {
		tmp = (((-sqrt(((((a_m * a_m) - sqrt(pow(a_m, 4.0))) / (y_45_scale * y_45_scale)) * (((((b_m * a_m) * (-4.0 * (b_m * a_m))) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * 2.0) * ((-a_m * a_m) * (b_m * b_m))))) / (4.0 * (b_m * a_m))) * (x_45_scale_m * y_45_scale)) / t_0) * (x_45_scale_m * y_45_scale);
	}
	return tmp;
}

Fortran

a_m =     private
b_m =     private
x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -a_m * b_m
    t_1 = (((a_m * b_m) * 4.0d0) / (y_45scale * x_45scale_m)) * (t_0 / (y_45scale * x_45scale_m))
    if (y_45scale <= 7.8d+105) then
        tmp = -sqrt((((2.0d0 * t_1) * ((b_m * a_m) * (b_m * -a_m))) * 0.0d0)) / t_1
    else
        tmp = (((-sqrt(((((a_m * a_m) - sqrt((a_m ** 4.0d0))) / (y_45scale * y_45scale)) * (((((b_m * a_m) * ((-4.0d0) * (b_m * a_m))) / (((y_45scale * y_45scale) * x_45scale_m) * x_45scale_m)) * 2.0d0) * ((-a_m * a_m) * (b_m * b_m))))) / (4.0d0 * (b_m * a_m))) * (x_45scale_m * y_45scale)) / t_0) * (x_45scale_m * y_45scale)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = -a_m * b_m;
	double t_1 = (((a_m * b_m) * 4.0) / (y_45_scale * x_45_scale_m)) * (t_0 / (y_45_scale * x_45_scale_m));
	double tmp;
	if (y_45_scale <= 7.8e+105) {
		tmp = -Math.sqrt((((2.0 * t_1) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_1;
	} else {
		tmp = (((-Math.sqrt(((((a_m * a_m) - Math.sqrt(Math.pow(a_m, 4.0))) / (y_45_scale * y_45_scale)) * (((((b_m * a_m) * (-4.0 * (b_m * a_m))) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * 2.0) * ((-a_m * a_m) * (b_m * b_m))))) / (4.0 * (b_m * a_m))) * (x_45_scale_m * y_45_scale)) / t_0) * (x_45_scale_m * y_45_scale);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	t_0 = -a_m * b_m
	t_1 = (((a_m * b_m) * 4.0) / (y_45_scale * x_45_scale_m)) * (t_0 / (y_45_scale * x_45_scale_m))
	tmp = 0
	if y_45_scale <= 7.8e+105:
		tmp = -math.sqrt((((2.0 * t_1) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_1
	else:
		tmp = (((-math.sqrt(((((a_m * a_m) - math.sqrt(math.pow(a_m, 4.0))) / (y_45_scale * y_45_scale)) * (((((b_m * a_m) * (-4.0 * (b_m * a_m))) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * 2.0) * ((-a_m * a_m) * (b_m * b_m))))) / (4.0 * (b_m * a_m))) * (x_45_scale_m * y_45_scale)) / t_0) * (x_45_scale_m * y_45_scale)
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(-a_m) * b_m)
	t_1 = Float64(Float64(Float64(Float64(a_m * b_m) * 4.0) / Float64(y_45_scale * x_45_scale_m)) * Float64(t_0 / Float64(y_45_scale * x_45_scale_m)))
	tmp = 0.0
	if (y_45_scale <= 7.8e+105)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * 0.0))) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(a_m * a_m) - sqrt((a_m ^ 4.0))) / Float64(y_45_scale * y_45_scale)) * Float64(Float64(Float64(Float64(Float64(b_m * a_m) * Float64(-4.0 * Float64(b_m * a_m))) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * 2.0) * Float64(Float64(Float64(-a_m) * a_m) * Float64(b_m * b_m)))))) / Float64(4.0 * Float64(b_m * a_m))) * Float64(x_45_scale_m * y_45_scale)) / t_0) * Float64(x_45_scale_m * y_45_scale));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = -a_m * b_m;
	t_1 = (((a_m * b_m) * 4.0) / (y_45_scale * x_45_scale_m)) * (t_0 / (y_45_scale * x_45_scale_m));
	tmp = 0.0;
	if (y_45_scale <= 7.8e+105)
		tmp = -sqrt((((2.0 * t_1) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_1;
	else
		tmp = (((-sqrt(((((a_m * a_m) - sqrt((a_m ^ 4.0))) / (y_45_scale * y_45_scale)) * (((((b_m * a_m) * (-4.0 * (b_m * a_m))) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * 2.0) * ((-a_m * a_m) * (b_m * b_m))))) / (4.0 * (b_m * a_m))) * (x_45_scale_m * y_45_scale)) / t_0) * (x_45_scale_m * y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[((-a$95$m) * b$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(a$95$m * b$95$m), $MachinePrecision] * 4.0), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 7.8e+105], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(-4.0 * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[((-a$95$m) * a$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(4.0 * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.79999999999999957e105

   1. Initial program 0.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 0.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(b \cdot a\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      14. pow2 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      15. times-frac N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale}} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      18. lower-/.f64 0.2

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot \left(-a\right)}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      20. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      21. lift-*.f64 0.2

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 0.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(b \cdot a\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}} \]

      14. pow2 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}} \]

      15. times-frac N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale}} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}} \]

      18. lower-/.f64 0.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot \left(-a\right)}}{y-scale \cdot x-scale}} \]

      20. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}} \]

      21. lift-*.f64 0.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}} \]

   8. Applied rewrites 0.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}}} \]

   9. Taylor expanded in y-scale around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{\color{blue}{{y-scale}^{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{\color{blue}{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       3. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       6. lower-pow.f64 4.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   11. Applied rewrites 4.4%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{\color{blue}{{y-scale}^{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   12. Taylor expanded in a around 0

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
   13. Step-by-step derivation

   14. Applied rewrites 9.6%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   if 7.79999999999999957e105 < y-scale

   1. Initial program 0.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 0.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(b \cdot a\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      14. pow2 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      15. times-frac N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale}} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      18. lower-/.f64 0.2

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot \left(-a\right)}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      20. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      21. lift-*.f64 0.2

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 0.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(b \cdot a\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}} \]

      14. pow2 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}} \]

      15. times-frac N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale}} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}} \]

      18. lower-/.f64 0.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot \left(-a\right)}}{y-scale \cdot x-scale}} \]

      20. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}} \]

      21. lift-*.f64 0.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}} \]

   8. Applied rewrites 0.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}}} \]

   9. Taylor expanded in y-scale around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{\color{blue}{{y-scale}^{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{\color{blue}{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       3. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       6. lower-pow.f64 4.4

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   11. Applied rewrites 4.4%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{\color{blue}{{y-scale}^{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   13. Applied rewrites 4.6%

       \[\leadsto \color{blue}{\frac{\frac{-\sqrt{\frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale} \cdot \left(\left(\frac{\left(b \cdot a\right) \cdot \left(-4 \cdot \left(b \cdot a\right)\right)}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot a\right) \cdot \left(b \cdot b\right)\right)\right)}}{4 \cdot \left(b \cdot a\right)} \cdot \left(x-scale \cdot y-scale\right)}{\left(-a\right) \cdot b} \cdot \left(x-scale \cdot y-scale\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 9.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{\left(a\_m \cdot b\_m\right) \cdot 4}{y-scale \cdot x-scale\_m} \cdot \frac{\left(-a\_m\right) \cdot b\_m}{y-scale \cdot x-scale\_m}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_0\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot 0}}{t\_0} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ (* (* a_m b_m) 4.0) (* y-scale x-scale_m))
          (/ (* (- a_m) b_m) (* y-scale x-scale_m)))))
   (/ (- (sqrt (* (* (* 2.0 t_0) (* (* b_m a_m) (* b_m (- a_m)))) 0.0))) t_0)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (((a_m * b_m) * 4.0) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	return -sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_0;
}

Fortran

a_m =     private
b_m =     private
x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = (((a_m * b_m) * 4.0d0) / (y_45scale * x_45scale_m)) * ((-a_m * b_m) / (y_45scale * x_45scale_m))
    code = -sqrt((((2.0d0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * 0.0d0)) / t_0
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (((a_m * b_m) * 4.0) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	return -Math.sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_0;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	t_0 = (((a_m * b_m) * 4.0) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m))
	return -math.sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_0

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(a_m * b_m) * 4.0) / Float64(y_45_scale * x_45_scale_m)) * Float64(Float64(Float64(-a_m) * b_m) / Float64(y_45_scale * x_45_scale_m)))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_0) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * 0.0))) / t_0)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = (((a_m * b_m) * 4.0) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	tmp = -sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * 0.0)) / t_0;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(N[(a$95$m * b$95$m), $MachinePrecision] * 4.0), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[((-a$95$m) * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 0.1%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

2. Taylor expanded in angle around 0

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 0.2%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(b \cdot a\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   12. *-commutative N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   14. pow2 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   15. times-frac N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   17. lower-/.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale}} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   18. lower-/.f64 0.2

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot \left(-a\right)}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   20. *-commutative N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   21. lift-*.f64 0.2

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

6. Applied rewrites 0.2%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{4 \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(b \cdot a\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(4 \cdot \color{blue}{\left(a \cdot b\right)}\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\color{blue}{\left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(b \cdot \left(-a\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}} \]

   12. *-commutative N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}} \]

   14. pow2 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(b \cdot \left(-a\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}} \]

   15. times-frac N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

   17. lower-/.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale}} \cdot \frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}} \]

   18. lower-/.f64 0.4

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{y-scale \cdot x-scale}}} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot \left(-a\right)}}{y-scale \cdot x-scale}} \]

   20. *-commutative N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}} \]

   21. lift-*.f64 0.4

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{y-scale \cdot x-scale}} \]

8. Applied rewrites 0.4%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) - \sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}}{\color{blue}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}}} \]

9. Taylor expanded in y-scale around 0

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{\color{blue}{{y-scale}^{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{\color{blue}{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

    2. lower--.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

    3. lower-pow.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

    4. lower-sqrt.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

    5. lower-pow.f64 N/A

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

    6. lower-pow.f64 4.4

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

11. Applied rewrites 4.4%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{\color{blue}{{y-scale}^{2}}}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

12. Taylor expanded in a around 0

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
13. Step-by-step derivation

14. Applied rewrites 9.6%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))